Translocation of a Polyelectrolyte through a Nanopore in the Presence of Trivalent Counterions: A Comparison with the Cases in Monovalent and Divalent Salt Solutions

Abstract

A polyelectrolyte threading through a nanopore in a trivalent salt solution is investigated by means of molecular dynamics simulations under a reflective wall boundary. By varying the chain length N and the strength E of the driving electric field applied inside the pore, the translocation time is carefully calculated to get rid of the bouncing effect because of the boundary. The results are analyzed under the scaling form ⟨τ⟩ ∼ N^αE^–δ and four driving force regimes; namely, the unbiased, the weakly driven, the strongly driven trumpet, and the strongly driven isoflux regime, are distinguished. The exponents are calculated in each regime and compared with the cases in the monovalent and divalent salt solutions. Owing to strong condensation of counter ions, the changes of the exponents in the force regimes are found to be nontrivial. A large increase in translocation time can be, however, achieved as the driving field is weak. The variations of the chain size, the ion condensation, and the effective chain charge show that the process is proceeded in a quasi-equilibrium way in the unbiased regime and deviated to exhibit strong nonequilibrium characteristics as E increases. Several astonishing scaling behaviors of the waiting time function, the translocation velocity, and the diffusion properties are discovered in the study. The results provide deep insights into the phenomena of polyelectrolyte translocation in various salt solutions at different driving forces.

1. Introduction

Biopolymer translocating across a membrane through a small pore exhibits several fascinating features and has attracted great attention from researchers for decades.¹⁻⁵ Because of spatial restriction at pores, translocation is proceeded accompanied by a huge re-arrangement of chain conformation to allow just few monomers to pass through the pore at a time.⁶ Owing to this unique feature, various mechanisms have been developed and devised at a pore to detect the species of the monomers passing instantaneously through it.^3,7 Today, molecular detection using nanopores has become a promising technique to sequence genetic codes of DNA molecules with ultra-high efficiency.^8,9

One of the main challenges encountered in nanopore sequencing is that the translocation velocity is usually fast.^7,10,11 It renders the detection of the genetic codes difficult, and thus the resolution of reading is low. Various methods have been proposed to slow down the process, for example, changing the solvent conditions (viscosity, pH, and temperature),^12,13 reducing the pore diameter,¹⁴ using different materials for membrane,¹⁵⁻¹⁷ modulating surface charges,^18,19 setting a salt concentration gradient,²⁰⁻²² introducing nanofiber mesh on top of the pore,^23,24 applying plasmonic fields to control pore resistance,^25,26 and so forth. By increasing the ionic strength of solutions, the translocation velocity has been shown to be reduced by several folds.²⁷⁻²⁹ Counter ion binding plays a crucial role on deciding the translocation time. For example, solutions with lithium ions exhibited a longer translocation time than the ones with sodium or potassium ions.²⁷ It is because the lithium ions have stronger binding ability with DNA molecules and the bound ions neutralize the chain charge, which slows down the process. Adding divalent counter ions provides a more efficient way to neutralize DNA molecules in the solutions and can even further reduce the translocation velocity.^30,31 It is known that the conformation and charge distribution of DNA molecules can be altered drastically by adding strong condensing agents such as trivalent or higher valent salt in solutions.³²⁻³⁷ Therefore, trivalent counter ions could be a more efficient agent to reduce translocation velocity. In this study, we perform molecular dynamics simulations to study DNA translocation in trivalent salt solutions. The results are further compared with the ones in monovalent and divalent salt solutions in order to understand the influences of the electrolytes of higher valence.

The starting of the theoretical study of polymer translocation can be referred to the pioneer work by Sung and Park in 1996.³⁸ Using a Fokker–Planck equation, they found that the translocation time τ scales as N^1+ν in the long chain limit and cross-overs to N^2+ν as N decreases below a threshold of about k_BT/Δμ, where N is the number of the monomers on the chain and Δμ is the chemical potential difference across the pore per monomer. For the case driven by an external force f, the translocation time is generally expressed by the scaling τ ∼ N^αf^–δ. At the early years, α = 1 and δ = 1 were predicted.³⁹ Later, Kantor and Kardar⁴⁰ set an upper bound of the scaling consistent with the prediction of Sung and Park: α = 1 + ν and δ = 1 with ν being the Flory exponent. Vocks et al.⁴¹ emphasized the importance of the memory effect because of the tension propagation on a chain and argued that α should be (1 + 2ν)/(1 + ν). Adopting the idea of tension propagation, Sakaue and co-workers⁴²⁻⁴⁷ found that translocation behavior is characterized by different scaling regimes. The latest proposition by them was given in ref (47) where the driving force of translocation was split into the unbiased (UB) regime, the weakly driven (WD) regime, and the strongly driven regime. The strongly driven regime was further divided into the trumpet and the stem-flower regimes. In the UB regime, α = 2 + ν was predicted; in the other three regimes, α was thought as 1 + ν. A similar two-value behavior was predicted for δ: zero in the UB regime and 1 in the other force regimes. Simulations, on the other hand, revealed that many factors can affect the scaling behavior of polymer translocation, for example, solvent quality,^48,49 viscosity,^50,51 temperature,⁵²⁻⁵⁴ pore size,⁵⁵⁻⁵⁷ interaction between the pore and the chain,⁵⁸⁻⁶⁰ confining geometry,⁶¹⁻⁶³ and so forth. Moreover, translocation phenomena can be mapped onto the first passage time problems.⁶ It has been recognized that the distribution of the first passage time can be very broad under certain conditions.^64,65 As a consequence, the calculation of the mean first passage time becomes difficult and cannot be easily obtained with good accuracy.^63,66 Owing to these reasons, the exponents reported in simulations are usually not consistent. The value of α is basically ranged between 1 and 1.9 for biased translocation and δ falls about in the range 0.79–1.25.^4,67

Recently, we have performed a series of molecular dynamics simulations and scaling analyses to investigate scaling behaviors of translocation of charged polymers through a small pore.⁶⁸⁻⁷⁰ Under a fixed-N condition, we investigated the problem in monovalent and divalent salt solutions, against the driving field strength E covering the entire force range. Four characteristic scaling behaviors were observed in the study of τ versus E. We were able to refine Sakaue’s scaling pictures⁴⁷ and separate the forces into four regimes, namely, the UB regime, the WD regime, the strongly driven trumpet (SDT) regime, and the strongly driven isoflux (SDI) regime.⁷⁰

In the UB regime, the driving electric field is negligibly weak. The translocation is effectuated merely via a random walk and, thus, the force exponent δ is zero. The driving force manifests its effect starting from the WD regime. The tension propagation theory is used to separate a translocation process into the propagation and the post-propagation stages. The translocation time is dominated by the post-propagation stage and predicted to vary with E^–1, which shows linear-response behavior. In the SDT regime, a series of tension blobs is formed from the pore entrance to the tension front on the cis side with the blob size being increased. The ensemble of the tension blobs looks like a trumpet and the translocation time is determined by the tension-propagation time, scaling as E^–δ_T with δ_T = p_z(1 – q). The simulations showed that δ_T is around 1.6 in the monovalent salt solutions.⁷⁰ Under an extremely strong driving force (i.e. in the SDI regime), the tension is very strong that the chain is driven through the pore in an isoflux way. The translocation time is thus inversely proportional to E.

The exponent α was studied by varying N at a fixed E and predicted to be α_I = 1 + ν_s in the SDI regime.⁷⁰ Decreasing E increased τ but the exponent was reduced to α_T = 1 + ν_s(1 – q) in the SDT regime which acquired a value of around 1.05. Furthermore, decreasing E turned to increase α when passing the WD regime, and finally, the system reached the UB regime with a scaling exponent α_U = 2 + ν_sz_p(1 – γ_p), significantly larger than 2.0.

It is difficult to determine α_W in the simulations but according to our scaling theory, the exponent should be α_U – 1. The values of the obtained exponents can be found in Table 1 in Section 3. The results showed that the phenomena of forced translocation are very complicated and essentially characterized by four scaling behaviors.

One of the astonishing findings in the previous study is that using monovalent or divalent salt in translocation does not significantly change the values of the exponents α and δ. We did observe that the translocation time was increased by a few folds in the WD regime in the divalent salt solution. However, the increasing entered in the problem mainly via a prefactor and gave only a small impact on the scaling exponents. It could be explained by the fact that divalent counter ions are, in fact, a weak condensing agent for charged polymers.^32,34,71 Only with a strong condensing agent such as trivalent salt or salt with higher valence, the conformation and charge distribution of polyelectrolyte chains can be changed drastically.^33,72,73

Recently, translocation of stiff polyelectrolytes through a long charged pore channel in salt solutions has been investigated by using a Poisson–Boltzmann mean-field theory in cooperation with a correlated electrolyte charge distribution.⁷⁴⁻⁷⁶ The “beyond mean-field theory” predicted several interesting phenomena in the presence of multivalent counter ions, including the reversals of the chain charge and chain electrophoresis,⁷⁴ the inversion of the ionic current in the channel,⁷⁵ the like-charge attraction between the chain and the pore,⁷⁷ the enhancement of the capture rate of polymer, and the extension of the translocation time because of the charge-inverted electroosmotic flow,^78,79 and so on. If the studied polyelectrolyte is much longer than the persistence length of the chain, the rigid rod approximation will not be valid because the fluctuations of chain conformation become important.⁷⁶ In this situation, tension propagation theories have been used to characterize the translocation dynamics and scaling behavior at different levels of driving forces.^47,70,76 However, these theories were developed under the framework of neutral polymers. It is thus important to know the limitation of the tension propagation theories. A direct investigation using long charged chains to study translocation behaviors through a thin membrane in different solution conditions is necessary in order to understand the impact of the electrostatic interaction and the valence of salt. On the basis of the above observations, we made the decision to simulate polyelectrolyte translocation in trivalent salt solutions in this study. The goal is to understand how the translocation time, the chain conformation, the ion distribution, and the drift-diffusion properties are affected by the presence of trivalent counter ions, in comparison with the cases with the monovalent and divalent ones, and elucidate the possibility in the reduction of the translocation velocity for future applications.

The paper is organized in the following way. The simulation model and setups are described in Section 2. The results are reported in Section 3. The studied topics include the mean translocation time (Sub-section 3.1), the variation of the chain size, and the changes of the ion condensation and the effective chain charge during a translocation process (Sub-sections 3.2 and 3.3). The waiting time function and the drift and diffusion properties of translocation are studied in Sub-sections 3.4 and 3.5. The discussions and conclusions are given in Section 4.

2. Simulation Model and Setup

Polyelectrolyte is modeled by a charged bead-spring chain in this study. We consider a “strongly charged chain” case here: each monomer on the chain is ionized and releases a cation into the solution. An amount of (Z:1)-salt molecules is added into the system. A salt molecule dissociates into one +Z-cation and Z monovalent anions in the solution. The excluded volume interaction of the monomer and the ion beads (or called the mobile beads together) is modeled by the Weeks–Chandler–Andersen potential⁸⁰

1

where ε_mm and σ_mm are, respectively, the interaction strength and the length parameter between two mobile beads. The chain is linearly connected by a harmonic potential

2

where k_sp is the spring constant, b is the bond length between two connected monomers, and b₀ is the equilibrium bond length. The charged beads interact mutually via the Coulomb interaction, expressed by

3

where Z_i and Z_j are the valences of the two beads and λ_B is the Bjerrum length which measures the electrostatic coupling distance between two unit charges in the solution with the Coulomb energy equal to the thermal energy k_BT. The temperature is denoted by T and k_B is the Boltzmann constant.

To study the translocation of charge chains, a membrane wall built with immobile wall beads is placed at the middle of the simulation box, which divides the space into two subspaces along the x̂-direction, called the cis and the trans regions. The two regions are connected by a pore channel passing through the membrane. The mobile beads interact with the wall beads via the excluded volume interaction taking the same form of eq 1 with ε_mw and σ_mw being the two interaction parameters. The simulation parameters are set to the following values: ε_mm = 1.2 k_BT, σ_mm = 1.0 σ, k_sp = 600.0 k_BT/σ², b₀ = 1.0 σ, λ_B = 3.0 σ, ε_mw = 2.5 k_BT, and σ_mw = 1.5 σ where σ is the length unit of simulation. The dimension of the simulation box is 200.0 σ × 48.0 σ × 49.36 σ. The radius of the pore is 2.25 σ and the length is 4.5 σ. Periodic boundary conditions are applied. The electrostatic interaction is calculated by using the particle–particle mesh Ewald method.^81,82 As explained in the Introduction section, we focus on the trivalent (Z = 3) salt case in this study. The amount of the added salt molecules is 256. The results are further compared with the monovalent (Z = 1) and the divalent (Z = 2) salt cases at the same concentration.

Initially, the head monomer of the chain is placed at the pore exit on the trans side with the chain body traversing the pore and staying mainly on the cis side. Langevin dynamics simulations are performed to equilibrate the system with the head monomer fixed. The damping time for temperature control is set to 1.0 t_u where is the simulation time unit. To simplify the model, we assume that the mobile beads have equal mass m. The number of monomers is varied from N_m = 8 to 384. Because the first five monomers are placed across the pore, the true number of monomers to be translocated from the cis side is N = N_m – 5. Translocation is started by switching on the electric field E⃗ = −Ex^ applied inside the pore and, at the same time, removing the immobilization restriction on the head monomer. The electric field drives the negatively charged beads, which are monomers and coions, in the pore toward the trans side while driving the positive ones, which are counter ions, toward the cis side. The field strength E is varied from 0.1 to 64.0 k_BT/(eσ), covering from a very weakly driving situation to a very strongly driving one. The bond stretching is small because the large spring constant k_sp is used. The average bond length ⟨b⟩ is 1.00(4) in the UB regime and 1.03(9) in the SDI regime.

Under a weakly driving condition, a process shall fail easily because the chain head can be retrieved back into the cis region by the entropic pulling of the chain body. Once it happens, the chain has to spend enormous time on searching for the pore entrance for the next trial of translocation. It is just impossible to deal with this kind of situation in simulations. To prevent failure, a reflective wall is set at the pore exit, visible only to the head monomer, to bounce it back into the trans region. The head monomer thus cannot enter the pore, and consequently, the failure of translocation is avoided.

We comment that, in real experiments or applications, such a reflective wall setting does not exist.^2,27,53 To handle the artifacts created by the reflective wall, we record the following three events in the simulations: (1) the last bouncing event of the head monomer from the reflective wall, (2) the first leaving event for the tail monomer to quit the cis region for the first time, and (3) the last leaving event to indicate the final leaving of the tail monomer from the cis region. The three events take place in turns in a process. The fact that the bouncing and leaving events can occur several times in a process is due to the diffusion nature of translocation. Because no more head monomer bouncing occurs after event 1, event 1 can be regarded approximately as the starting point of a real translocation without the reflective wall. We define the translocation time τ to be the duration between event 1 and event 2. This definition gets rid of the artifact of the reflective wall and the result is expected to be close to the real translocation. To elucidate the impact of the reflective wall, the escape time, τ_e, which is the time spent over the entire process since the switching-on of the driving field in the pore to the final leaving of the chain (the event 3), is measured in parallel and compared with the translocation time. Two other time scales are also calculated: the bouncing time, τ_b, which is the elapsed time since the switching-on of the field to event 1, and the leaving time, , which is the time difference between event 2 and event 3. We remark that the timing for τ is started at the last bouncing of the chain and stopped at the first leaving, and thus, the leaving time is not involved in the translocation time in this study. This choice is used to have the most likelihood to observe the time-reversible characteristics in a quasi-equilibrium process. We will show later that is negligibly small in the long chain limit in comparison with τ. Hence, whether is involved in the translocation time or not will not alter the scaling behavior. Please notice that the time spent for the chain tail monomer to traverse the pore channel after event 3 is not accounted for in the calculation of τ_e. It is because the extent depends on the pore length and the interaction with the membrane wall. We do not account for this part of the time so that the escaping of a chain out of the cis region can be studied properly from the theoretical point of view.

For each studied pair of parameters (N, E), 500 independent translocation events are collected. The data are then calculated by using the standard statistical analysis. More information about the modeling and settings of simulation can be found in our previous paper.⁶⁸

In this work, the hydrodynamic interaction is not considered. We study translocation behavior of long polyelectrolyte chains through a short pore channel. Because the pore radius is very small, comparable to the monomer and ion sizes, the formation of the electrical double layer inside the pore channel is expected to be not important. The ionization of the membrane surface in the solution is thus not modeled and, consequently, the electro-osmotic flow is not involved in the study. We do not expect that the electro-osmotic flow can have a great influence on the translocation behavior. It is because the flow created prior to the entrance of the chain shall be rapidly blocked out by the chain, especially when the chain is long.

We stress that the main purpose of this work is to establish a clear scaling picture of polyelectrolyte translocation over the entire driving force space. The studied model is chosen to be as simple as possible to permit pinning-down the essential physics for translocation. The obtained results, generally, facilely serve as references for experiments. Other simulation parameters such as the ion size, the ion mass, the chain stiffness, the pore diameter and pore length, the surface properties, and so forth are also important and could have their own influences on the translocation. However, we do not expect a large modification of the scaling behavior by these factors in viewing that the experimental data obtained from different DNA translocation systems using different ions lie compatibly with our simulations, as shown in Figure 15. Nonetheless, the influences of these factors deserve more detailed investigations in the future.

Figure 15.

Simulation results, ⟨τ⟩ vs N, in comparison with the experimental ones. The simulation data have been converted to the real physical units: time is ms, electric field strength in mV/nm, and N in number of basepairs. The parameters Z and E can be read in the legend. The experimental data are taken from ref (27) (symbols: tilted square open, circle open), ref (30) (symbols: triangle dotted, triangle down open, pink triangle up solid), ref (31) (symbols: box, pink circle solid), ref (28) (symbol: pentagon open), ref (29) (symbol: plus), and ref (22) (symbol: multiplication), where the open symbols denote the experiments done in monovalent salt solutions and the solid symbols (in magenta color) the ones in divalent salt solutions. The black dashed line is an extension of the UB regime from the simulations. The green dashed one indicates the rough position of the SDT regime. Above it, we are in the WD regime. Below is the SDI regime.

To illustrate a translocation process, snapshots of simulation are given in Figure 1. The chain comprises 128 monomers driven by an electric field E = 1.0 k_BT/(eσ) applied inside the pore. The number printed near the top-left corner of a frame indicates the progress of the process, defined as the ratio of the elapsed time, t, to the process time, τ_e. In the snapshots, the connected yellow beads are the chain monomers. The red and white beads represent the trivalent and monovalent counter ions, respectively. The coions are colored in green. We can see that a considerable number of the trivalent counter ions are condensed on the chain. Some of them are even dragged across the pore with the chain.

Figure 1.

Snapshots of a translocation process. The connected yellow beads represent the chain. The trivalent counter ions [(+3)-ions], monovalent counter ions [(+1)-ions], and coions [(−1)-ions] are colored in red, white, and green, respectively. A hallow wall divides the space into the cis and trans regions, with a pore channel set at the center. The driving electric field E⃗ = −1.0x^ is applied inside the pore. The number of the monomers is 128. The progress of translocation is indicated at the top-left corner of each snapshot. The chain is gradually transported from the cis side (on the left) to trans side (on the right) by the driving field. To visualize the chain inside the pore, the wall beads (in gray color) have been drawn with a certain degree of transparency.

In the following text, we will report physical quantities using the simulation units: σ for length, m for mass, k_BT for energy, and e for charge. To shorten the notation, the unit of a physical quantity will not be given explicitly. For example, the field strength “E = 0.2” in the text means E = 0.2 k_BT/(eσ) and the translocation time “τ = 100” means .

Because of the limitation of the computing power in this study, the scaling behaviors investigated in certain force regimes may not have a large enough extension with the data range spanning more than one decade. Hence, the exponents reported from these small regimes should be interpreted with care. It should be regarded as an indication for a scaling trend from a strict point of view, where the translocation likely shows such a power-law dependence in the regime. Here, we do our best to report the most reliable data from the simulations. The scaling exponents are calculated via the study of the trend of variations against E and N (refer to Supporting Information, Figure S3). The results are also compared across the different salt cases to make sure for the existence of a scaling dependence.

3. Results and Discussions

3.1. Mean Translocation Time, Escape Time, Bouncing Time, and Leaving Time

We study the scaling behavior of the mean translocation time in the trivalent salt solution under the form: ⟨τ⟩ ∼ N^αE^–δ. The mean escape time ⟨τ_e⟩ is also calculated for comparison.

Here, N denotes the number of monomers in the cis region at the starting of the translocation. The five monomers initially placed across the pore channel do not account for it. Therefore, what we study in this work is the scaling in the main part of translocation concerning the proper transportation of the N monomers out of the cis region. In a real translocation process, there are still two additional times involved: the first one is the time spent for the head monomer to traverse the pore channel from the entrance into the pore and the second is the time required for the tail monomer to travel across the pore before entering the trans region. The two times are influenced by the pore properties, such as the pore diameter, the pore length, the surface friction, and so forth, and are expected to possess their own scalings which could perturb the scaling study for the translocation time. Thus, the translocation time calculated here does not include the two times.

We recall that the timing for τ is started at the last bounce of the head monomer to the reflective wall and ended at the first leaving of the chain from the cis region in this study. The timing for τ_e is, on the contrary, started at the beginning of the process and ended at the last leaving from the cis region. The results are presented in Figure 2a, as a function of E, in a log-log plot for various chain lengths. The ⟨τ⟩ data are connected by thick line segments while the ⟨τ_e⟩ ones are connected by thin segments for distinction.

Figure 2.

(a) Mean translocation time ⟨τ⟩ (thick lines) and mean escape time ⟨τ_e⟩ (thin lines) vs the strength of the driving field E in a log-log plot. The number of monomers N for each curve is fixed and the value can be read in the legend. (b) Mean bouncing time ⟨τ_b⟩ (in the upper panel) and mean leaving time (in the lower panel) vs E for different N values.

We can see that both ⟨τ⟩ and ⟨τ_e⟩ are about constant at small E. It shows that the translocation process is essentially not affected by the driving field, and thus, the system is situated in the so-called UB regime. The difference between ⟨τ⟩ and ⟨τ_e⟩ is, however, very large, showing the difficulty in a “direct” translocation (without bouncing) occurring in the UB regime. Therefore, we should be cautious in using the ⟨τ_e⟩ data because enormous time (which is τ_b) is spent on the trials to overcome the entropic barrier of translocation. The translocation thus should be studied by ⟨τ⟩, and not by ⟨τ_e⟩, because large ⟨τ_b⟩ can deviate the scaling behavior.

With increasing E over some threshold, ⟨τ⟩ turns to decrease its value. The decreasing rate is fast at the beginning. It then slows down and reaches a scaling regime with the decreasing exponent of δ ≃ 1.62. The exponent finally attains a value of around 1 under the extremely strong driving and ⟨τ⟩ becomes inversely proportional to E. The observed three behaviors can be mapped onto the WD, SDT, and SDI force regimes, as discussed in the Introduction section.

Different to the previous studies in the monovalent salt solution^68,70 (refer also to the Supporting Information, Figure S1), the translocation exhibits a nonlinear variation in the WD regime in the trivalent salt solution, with a decreasing ⟨τ⟩ faster than the expected linear response behavior.

This nonlinear variation results from the strong condensation of the counter ions which collapse the charged chain into a compact globule^34,71,83 in the cis region. It largely increases the difficulty in proceeding a translocation. To succeed the threading, the chain should be “detached” from the collapsed state near the pore entrance via the decondensation of the trivalent counter ions. The process is realized by thermal fluctuations in the UB regime, and therefore requires enormous time for trials. Increasing E beyond a threshold enables the shearing of the bindings of the condensed counter ions in a nonlinear way, where the chain is forced to enter to the pore. As a result, the translocation time drops rapidly in the WD regime, deviating from the linear expectation. The nonlinear decreasing behavior is also observed in the divalent salt solution using the newly defined translocation time (refer to Supporting Information, Figure S2). When the strength of the filed E goes into the strongly driving regime, the bindings due to the condensation of the counter ions are no longer an effective hurdle for the translocation. The scaling thus shows characteristic SDT and SDI behaviors similar to the ones in the monovalent salt solution.

The escape time ⟨τ_e⟩ approaches ⟨τ⟩ rapidly as E increases. The difference ⟨τ_b⟩ is calculated and plotted in the upper panel of Figure 2b. We can see that when E is small, ⟨τ_b⟩ dominates the escape time. It is thus important to discount this artificial time from ⟨τ_e⟩ to correctly study the translocation without bouncing. In the log-log plot, ⟨τ_b⟩ exhibits four astonishing scaling behaviors, E⁰, E^–8.0, E^–1.6, and E^–5.7, indicating that the forced translocation faces four different resistances. The four scalings are found corresponding exactly to the UB, WD, SDT, and SDI force regimes. This is an alternative evidence for the existence of the four force regimes, consistent with the pictures depicted by the tension propagation theory.⁷⁰ It is worth noticing that the ⟨τ_b⟩ curves for different N turn to follow a common track of variation when the driving force enters the WD regime.

The leaving time, , which is the time difference between the first and the last leaving events of the chain, is plotted in the lower panel of Figure 2b. We observe that is about a constant in the UB and WD regimes, independent of the chain length. It turns to exhibit a E^–7.7 decreasing in the strongly driving regimes. Noticeably, the value of is negligibly small in comparison with ⟨τ⟩. Therefore, the scaling of the translocation time shall not be altered whether is considered as a part of the translocation time or not.

Figure 3a,b presents the variations of ⟨τ⟩, ⟨τ_e⟩, ⟨τ_b⟩, and against N at various driving electric fields. For large N, three scaling behaviors for ⟨τ⟩ are identified under the negligibly-weak, strong, and extremely-strong driving conditions. They are N^2.41, N^0.92, and N^1.25, respectively, obtained by the least-square fit method from the last three data points. It permits us to determine the exponents α_U, α_T, and α_I in the corresponding UB, SDT, and SDI force regimes. The obtained α_T value is smaller than α_I. It may look strange at the first glance and seems to indicate the occurrence of a possible nonphysical scenario: the extrapolation of the ⟨τ⟩ curve for large E (such as 64.0) shall eventually intercept the one for small E (such as 1.5) at a large enough N, and thus the obtained nonphysical result is that a weaker driving field leads to a faster translocation. We stress that this kind of situation will not occur. With increasing N at a fixed driving field, the extension of the time curve will not stay always be in the SDT regime but will switch to the SDI one at some moment. As a result, the curves at the large N will become in parallel with each other with the scaling exponent equal to α_I, as shown in Figure 2b of ref (70). For N smaller than 10, our simulations show that the ⟨τ⟩ curves deflect inward for different driving forces. It is due to the finite size effect of the chain length which leads to the rapid convergence behavior.

Figure 3.

(a) ⟨τ⟩ vs N (thick lines) and ⟨τ_e⟩ vs N (thin lines) at various driving electric fields. The field strength E can be read in the legend. (b) ⟨τ_b⟩ vs N (in the upper panel) and vs N (in the lower panel) at different E.

We remark that the scaling exponent calculated from the escape time ⟨τ_e⟩ at very weak fields is significantly larger than 2.41 and is not a good value for α_U. It is because ⟨τ_e⟩ is dominated by the bouncing time ⟨τ_b⟩ in this regime, which exhibits a stronger scaling N^3.2 as shown in the upper panel of Figure 3b. The leaving time of the chain tail is plotted in the lower panel of the figure. We can see that the value is essentially independent of N and significantly smaller than ⟨τ⟩.

The obtained δ and α exponents are listed in Table 1. The ones for the monovalent and divalent cases are given beside for comparison. The exponents in the monovalent and divalent salt solutions have been re-calculated by using the newly defined translocation time ⟨τ⟩, which discounts the bouncing stage and the leaving stage. The plots of ⟨τ⟩, ⟨τ_e⟩, ⟨τ_b⟩, and versus E and versus N in the two salt solutions have been given in Figures S1 and S2 of the Supporting Information for reference. The variations of the exponents calculated from every pair of the consecutive data along the ⟨τ⟩ curves are presented in Figure S3. The results show that δ is really zero in the UB regime and reaches a plateau value, around 1, in the WD regime if the studied case is in the monovalent salt solution. In the divalent salt solution, the δ value increases smoothly in the WD regime, whereas in the trivalent one, it increases abruptly. The exponent then attains a higher plateau value in the SDT regime and finally decreases to a asymptotic value 1 in the SDI regime for the three studied salt cases. The variation of α with increasing N (refer to Figure S3b) reveals that there exist three important plateau values, corresponding to the characteristic scaling behaviors in the UB, SDT, and SDI regimes. From these plateaus, we are able to determine the values for δ and α in the four force regimes.

Table 1. List of the Exponents δ and α in the UB, WD, SDT, and SDI Regimes for the Monovalent (Z = 1), Divalent (Z = 2), and Trivalent (Z = 3) Salt cases^a.

regime	exponent	Z = 1	Z = 2	Z = 3
UB	δU	0.0	0.0	0.0
WD	δW	1.02(8)
SDT	δT	1.55(11)	1.58(11)	1.62(14)
SDI	δI	1.02(3)	0.99(6)	0.95(6)
UB	αU	2.73(15)	2.46(23)	2.41(33)
WD	αW
SDT	αT	1.06(1)	1.06(3)	0.92(9)
SDI	αI	1.38(4)	1.47(10)	1.25(7)

^aThe em dash “—” is used to denote the situation that the scaling exponent cannot be defined or measured from the simulations

In Table 1, we can see that δ_T is about 1.6 and δ_I is about 1.0 for the three salt cases. For δ_W, only the Z = 1 case is reported because the other two cases do not exhibit a clear scaling variation in the regime, as mentioned above. δ_W is about 1.0 at Z = 1, which follows the expected linear response behavior under a weak driving condition. The exponent α_U attains a value of 2.73(15), 2.46(23), and 2.41(33) for Z = 1, 2, 3, respectively. It shows an unexpected decreasing behavior with the salt valence. As a consequence, the translocation process is not slowed down in a salt solution with higher valence. This is a counter-intuitive result and will be explained later. The exponent α_W shall not be reported in the table because there appears no remarkable plateau in the corresponding WD regime, as shown in Figure S3b. In the SDT regime, the α_T exponent attains a value of around 1.0. For the even stronger isoflux driving, exponent α_I is found around 1.3–1.5, depending on the salt valence.

To illuminate the effect of salt valence, we present, in Figure 4, the plots of ⟨τ⟩ versus E at N = 27, 123, 379 and ⟨τ⟩ versus N at fixed E = 0.1, 1.0, and 64.0 in the monovalent, divalent, and trivalent salt solutions. A notable difference for the trivalent case, as shown in Figure 4a, is the nonlinear decreasing behavior in the WD regime. The divalent case also reveals a similar, but weaker, nonlinear decreasing behavior, compared to the linear one for the monovalent case. An astonishing finding is that the translocation time becomes shorter in the UB regime as the salt valence increases. For example, ⟨τ⟩ is 1.11 × 10⁵, 2.99 × 10⁴, and 1.59 × 10⁴, respectively, for the three salt cases as N = 123. What has happened exactly, since we waited for a stronger resistance to slow down the process in a solution with higher salt valence? To answer the question, we have studied the chain size in the three salt solutions. The mean radius of gyration of the chain on the cis side, ⟨R_g,C⟩, is 15.1, 6.32, and 4.46, respectively, before proceeding a translocation process. Translocation in the UB regime is expected to effectuate via a normal diffusion process with the diffusion coefficient D of about ⟨R_g,C⟩²/⟨τ⟩, which gives the three values, 2.05 × 10^–3, 1.34 × 10^–3, and 1.25 × 10^–3, respectively. Therefore, the resistance (which is proportional to 1/D) is indeed stronger in the trivalent salt solution. However, the size of the chain is diminished by the condensation of the multivalent counter ions, which reduces the required diffusion distance for translocation. As a result, the translocation time is shortened for the trivalent salt.

Figure 4.

(a) ⟨τ⟩ vs E at the three chain lengths, N = 27 (solid line), 123 (dashed line), and 379 (solid line); (b) ⟨τ⟩ vs N at the three field strengths, E = 0.1 (solid line), 1.0 (dashed line), and 64.0 (solid line). The salt valence Z can be read in the legend.

In the WD regime, we observe that the value of ⟨τ⟩ becomes significantly larger for Z = 3 because of the nonlinear response behavior. The regime is, hence, the optimal regime being searched for slowing down the process in real experiments or applications. In the SDT and SDI regimes, the time curves decrease in a similar way for the three salt cases, showing that the process is strongly dominated by the driving forces. Figure 4b confirms again that ⟨τ⟩ is larger with Z = 3 in the WD regime (E = 1.0). At the weaker and the stronger fields, the trivalent salt possesses a smaller translocation time.

3.2. Variation of the Chain Size

The variation of the chain size during a translocation process is studied in this section. We use the mean radius of gyration to characterize the chain size in a region X, defined by

4

where r_i is the position of the monomer i and r_cm,X is the center of mass of the N_X monomers in the region. We are interested in understanding the size variation in the cis region (X = C), the trans region (X = T), and the whole region (X = A). The variation is studied against the scaled translocation coordinate s̃ = s/N with s being the number of the monomers on the trans side. The value s̃ denotes the progress of a process. Figure 5 presents the chain size variations at five driving electric fields in the trivalent salt solution with N = 123.

Figure 5.

Mean radius of gyration of the chain segments in the whole space (⟨R_g,A⟩), the cis region (⟨R_g,C⟩), and the trans region (⟨R_g,T⟩), against the scaled translocation coordinate s̃ at the driving electric fields E = 0.1, 2.0, 4.0, 8.0, and 64.0. N is 123 and the salt valence Z is 3. The dashed line in cyan color in Panel E = 0.1 is a parabolic curve by fitting the ⟨R_g,A⟩ data.

During a translocation process, the chain is segregated by the membrane wall into two portions. The degree of segregation can be measured by ⟨R_g,A⟩. In the UB regime (refer to Panel E = 0.1 in the figure), the value increases with advancing the process up to s̃ = 0.5. After this, the large portion of the chain switches from the cis side to the trans side, resulting in the reduction of the chain size.

The entire ⟨R_g,A⟩ curve looks symmetric to the point s̃ = 0.5, which is a characteristic of time reversibility. It shows that the process is performed under a quasi-equilibrium condition. We discover that the data curve is approximately parabolic. The fitting curve has been plotted on top of the data, in a cyan dashed line, for comparison. This time-reversible characteristic can be also seen from the chain size variations in the cis and trans regions. ⟨R_g,C⟩ decreases monotonically with s̃ while ⟨R_g,T⟩ increases. The two curves are in mirror image with the other.

Increasing the driving field to be in the WD (at E = 2.0), SDT (at E = 4.0 and 8.0), and SDI (at E = 64.0) regimes skews the ⟨R_g,A⟩ curve to the left-hand side. The stronger the field strength, the higher the peak of the curve, but the lifting at s̃ = 1.0 does not show a monotonic increase with E. We observe that the decreasing of ⟨R_g,C⟩ turns to be more retarded and angled. On the other hand, the ⟨R_g,T⟩ curve grows and approaches the ⟨R_g,A⟩ near the end of the process.

We comment that the value of ⟨R_g,C⟩ is smaller than ⟨R_g,A⟩ at s̃ = 0 because the first five monomers are placed in the pore channel at the starting point. These monomers are not included in the calculation of ⟨R_g,C⟩, which causes the difference. Similarly, few monomers stay still in the pore channel right at the moment s̃ = 1.0, and therefore, ⟨R_g,T⟩ < ⟨R_g,A⟩. The difference between ⟨R_g,T⟩ and ⟨R_g,A⟩ at s̃ = 1.0 decreases with increasing E. It is because under the strong driving force, the chain on the trans side does not have enough time to relax and is crowded near the membrane wall and forms a pancake-like structure. As a result, the mean distances of the monomers to the center of mass in the two calculations have no big difference.

A comparison between the salt valences for the variation of ⟨R_g,A⟩ is given in Figure 6. We can see that the chain size is significantly large in the monovalent salt solution. Similar to the trivalent salt, the ⟨R_g,A⟩ curve is symmetric with respect to the midline at the weak field E = 0.1 and can be fitted by a parabola. Increasing E augments the peak height and skews the curves. Different to the trivalent case, the curves are skewed to the right-hand side and the final value monotonically decreases. For the divalent salt case, a nonmonotonic behavior, increasing first and decreasing later, is observed at s̃ = 1.0.

Figure 6.

Variations of ⟨R_g,A⟩ at different driving fields E in the (a) monovalent, (b) divalent, and (c) trivalent salt solutions, for N = 123.

The physics are explained below. Under a fast driving, the chain segments translocated into the trans region can be crowded near the pore exit and do not have enough time to be relaxed into the trans space. Therefore, the value of ⟨R_g,A⟩ at s̃ = 1 decreases with increasing E, as seen in the monovalent salt solutions. A second effect comes in for the divalent and trivalent salt cases, related to the de-condensation of counter ions when the chain passes through the pore. The size of the chain portion arrived on the trans side can hence become larger than the one in an equilibrium condition because the divalent or the trivalent counter ions need time to condense back onto the chain portion, which reduces the chain size. The interplay between the above two effects results in the nonmonotonic behavior. How the number of the condensed counter ions varies with the driving electric field in a translocation process will be studied in the next section.

Snapshots for N = 123 at E = 1.0 (in the WD regime) are presented in Figure 7 for the three studied salt cases. In the left panels are those at 10% of the translocation process, whereas in the right panels the ones at 90% of the process. We can see that the chain portions are remarkably collapsed on both sides of the membrane because of the condensation of the divalent and trivalent counter ions. The trivalent counter ions can collapse the chain in a more compact way.

Figure 7.

Snapshots of translocation for N = 123 at E = 1.0 in the (Z:1)-salt solutions, at 10% (left panel) and 90% (right panel) of the process. The yellow and green beads represent the monomers and the coions, respectively. The white beads are the monovalent counter ions. The red ones represent the divalent counter ions for the divalent salt case and the trivalent counter ions for the trivalent case.

While the condensation of counter ions increases the difficulty in threading the chain through the pore, the reduction of the chain size in the space decreases the time needed for a translocation. The competitions between these effects and the nonlinear response in the WD regime (discussed in the previous section) result in a prolonged translocation in the trivalent salt solution, as shown in Figure 4a at E = 1.0.

3.3. Ion Condensation and Effective Chain Charge

According to Manning’s theory,^84,85 ion condensation can take place on a charged chain as the line charge density of the chain is larger than the critical value ρ_* = e/λ_B in a solution, to reduce the line charge density to be the critical value. The condensation occurs in the neighborhood of the chain and is examined in a cylindrical region surrounding the chain backbone with the cross-section radius equal to λ_B.^37,73,86,87 We study the number of the ions presented in the region. The results are given in Figure 8 where two sets of the curves are plotted: ⟨N_+Z,C^(c)⟩, the mean number of the condensed (+Z)-counter ions in the cis region, and ⟨N_+Z,T⟩, the one in the trans region.

Figure 8.

Mean number of the condensed (+Z)-ions on the cis side, ⟨N_+Z,C^(c)⟩, and on the trans side ⟨N_+Z,T^(c)⟩ vs the scaled translocation coordinate s̃ for N = 123 in the (a) monovalent, (b) divalent, and (c) trivalent salt solutions. The ⟨N_+Z,C⟩ curves decrease with s̃ whereas the ⟨N_+Z,T^(c)⟩ increase with s̃. The field strength E can be read in the legend of Panel (c).

We anticipated to see a decreasing variation for ⟨N_+Z,C^(c)⟩ and an increasing variation for ⟨N_+Z,T⟩ because the amount of the condensed ions is directly related to the number of the monomers, which decreases on the cis side and increases on the trans side. This is exactly what we see in the figure. At the weak driving field E = 0.1, the translocation is proceeded in a quasi-equilibrium manner where the decreasing and increasing behaviors are identical with the reversal of the process. Noticeably, the two behaviors are linear against s̃, asserting that the number of the condensed counter ions is proportional to the number of the monomers on each side.

Increasing E slightly moves the ⟨N_+Z,C^(c)⟩ curve upward but decreases drastically ⟨N_+Z,T⟩ by reducing the slope of the curve. It is because when a process is proceeded quickly, the de-condensed counter ions do not have sufficient time to be relaxed on the cis side and go away from the chain, and thus, ions are found cumulative in the vicinity of the chain. Similarly, insufficient time is given for the counter ions to condense back onto the chain portion in the trans region. As a result, fewer condensed counter ions are found, compared to the situation in the quasi-equilibrium conditions. Because the required time for the condensation of a counter ion is much longer than the time for the de-condensation, the change on the set of the ⟨N_+Z,T^(c)⟩ curves is more drastic. Nonlinear variations occur as E is large for both ⟨N_+Z,C⟩ and ⟨N_+Z,T^(c)⟩, particularly on the cis side for Z = 2 and Z = 3, and on the trans side for Z = 1.

The effective chain charge Q_eff can be computed by summing the charges of all the condensed ions and the chain. The scaled mean effective charge ⟨Q̃_eff⟩ = ⟨Q_eff⟩/N are plotted in Figure 9 against s̃ for the three studied salt cases.

Figure 9.

Scaled mean effective chain charge ⟨Q̃_eff⟩ vs s̃ in the (a) monovalent, (b) divalent, and (c) trivalent salt solutions with N = 123. The strength of the driving electric field is given in the legend of (c).

We observe that ⟨Q̃_eff⟩ keeps at about a constant in the translocation at E = 0.1, showing again that the process was run in quasi-equilibrium. For Z = 1, the effective chain charge is about 24% of the bare chain charge. The value is somewhat lower than the Manning’s value, σ/λ_B = 0.33, which is predicted for a rigid chain in a bulk solution. Here, the studied chain is flexible. In the divalent salt solution, the chain is nearly neutralized whereas in the trivalent one, the effective chain charge is positive, showing an overcharging of the chain by the condensed counter ions.

Increasing E pushes the translocation process away from the equilibrium situation. As explained above, the responding time is not long enough to allow the ions to condense back onto the chain completely in the trans region. Therefore, the chain charge becomes more and more negative. The stronger the driving field, the larger the reduction of the effective charge, and the curve shows more effect of nonlinearity.

We remark that the condensed counter ions are not firmly bound to the chain. They can move on the surface of the chain by shearing of the electric field when the chain passes through the pore. They can be kicked off the chain simply by the thermal fluctuations or by the brutal force of a strong electric field, which tears off the condensed ions from the chain in the SDT or SDI regimes because the energy of condensation is few k_BT (about λ_BZk_BT/σ). The other counter ions then condense back onto the chain to maintain a dynamic decondensation–condensation equilibrium. Therefore, although overcharged in the trivalent salt solution in the weak field, the negatively charged chain segments can be still driven, monomer by monomer, toward the trans side inside the pore, whereas the condensed counter ions are sheared toward the cis side, following the direction of the electric field.

3.4. Waiting Time and Drift Velocity

To understand the dynamics of translocation, the waiting time function W(s) is calculated. The function describes the dwelling time at each state s of the translocation coordinate. We recall that a state s can be revisited several times in a process because of the diffusion nature of translocation. W(s) is hence an accumulation of the time at the state s which the chain stays with. Two kinds of waiting time functions are studied: the first is the waiting time calculated from the starting of the process until the last leaving of the chain from the cis region and the second is the one calculated since the last bouncing of the head monomer by the reflective wall up to the first leaving event of the chain. The latter excludes the enormous time duration spent in overcoming the entropic barrier at the beginning and is expected to be close to the waiting time function observed in a real experiment. We denote it by W(s). The previous one involves the bouncing stage and the leaving stage and is denoted by W_e(s) for distinction. The waiting time functions are related to the mean translocation time via the integrals

5

The difference between W_e(s) and W(s) gives the “spectrum” of the bouncing and leaving time over the state s.

The two kinds of waiting time function for N = 123 in the trivalent salt solution are presented in Figure 10 in a semi-log plot at different driving fields. The results for the monovalent and divalent salt cases are plotted beside for comparison.

Figure 10.

Waiting time W vs normalized translocation coordinate s̃ (in colored lines) and W_e vs s̃ (in black, dashed or dotted lines) in the (a) monovalent, (b) divalent, and (c) trivalent salt solutions. The number of monomer N is 123. The driving field strength E is given in the legends.

In the UB regime (E ≤ 0.01 for Z = 1, E ≤ 0.1 for Z = 2, and E ≤ 0.5 for Z = 3), the shape of the W function looks like an inverted letter U and is symmetric to the middle of the process at s̃ = 0.5. It ascertains the correctness of using the newly defined translocation time in the study because the waiting time function should be identical to itself by reversing the time arrow if a process is done under quasi-equilibrium conditions. For the corresponding W_e curve, we observe a huge and broad peak appearing at the starting of the process, which breaks the symmetry of the curve. It shows that enormous time is required to surpass the entropic barrier of translocation. Therefore, a successful translocation via just a single trial of threading is nearly unlikely in a real experiment in the UB regime. In this study, a reflective wall has been introduced to bounce the head monomer back into the trans region. It permits the chain to continuously do the trials for the translocation. The translocation time and the waiting time function were then studied by skipping the bouncing stage to extract the correct duration in the UB regime.

Increasing E shifts the W and W_e curves downward. It follows the intuition that a stronger driving field shall shorten the dwelling time. The peak of the W_e curve at small s̃ diminishes and narrows down quickly. Beyond E = 1.0 for Z = 1 and E = 2.0 for Z = 2, 3, the peak disappears and the W_e curve turns to be approximately the W curve. As E ≥ 16.0, the peak grows back weakly. It can be attributed to the inertia effect of the chain in response to the switching-on of the driving field, which drags the chain and thus increases the dwell time near the starting point.

Originally in the UB regime, the W curve exhibits a rounded hump in the mid-range. The hump is flattened with increasing E and becomes a plateau. The plateau then tilts to the left side under stronger driving conditions and a new hump appears on the right side at about s̃ = 0.7, 0.8, and 0.9 for the three salt cases, respectively. The higher the salt valence, the later the appearance of the hump in a process.

Figure 11 presents the waiting time function at the bouncing and leaving stages. The bouncing stage concerning the bouncing back of the head monomer is due to the introduction of the reflective wall boundary in the study. The leaving stage is related to the leaving of the chain end from the cis region, which is issued from the diffusive nature of a translocation. The two dwelling time functions, W_b(s) and , can be calculated at each state s of the two stages. The W_b(s) function shows its importance near the starting of a process, while the one near the end of the process. Therefore, we are able to distinguish them when the two curves are summed together in the figure.

Figure 11.

Waiting time function W_b + vs s̃ in the (a) monovalent, (b) divalent, and (c) trivalent salt solutions for N = 123. The driving field strength E is given in the legend.

We find that W_b(s), the left branch of the curve, shows its domination over the dwelling time W(s) when the driving force is in the UB and WD regimes. The range of the influence is quite broad and can be even up to about 80% of a process. How the W_b(s) curve diminishes with E and the range of the influence reduces can be clearly visualized in the figure. The right branch of the curve is contributed from the function. Its value is several orders smaller and has no real impact on the behavior of W(s) in the scaling scale. The width of shrinks with E too. Beyond some driving field, the two branches separate completely.

The drift velocity shows how fast a polymer chain threads through a pore in a translocation process. It can be calculated by V(s) = σds/dt ≃ σΔs/Δt = σ/W(s).

If the bouncing and leaving stages are accounted for in the study, the drift velocity should be V_e = σ/W_e(s). We discovered that the two velocities show interesting scaling behaviors if they are plotted against the number of the monomers in the cis region, denoted by m, rather than the one in the trans region (which is the translocation coordinate s). The results, V versus m (in colored lines) and V_e versus m (in black, dashed or dotted lines), are presented in Figure 12.

Figure 12.

Mean translocation velocities V (in colored lines) and V_e (in black, dashed or dotted lines) as a function of the number m of the monomers in the cis region in the (a) monovalent, (b) divalent, and (c) trivalent salt solutions. The chain has N = 123. The driving field strength E for V is given on the right side of the curve. The one for V_e is given in the legend.

The V curve decreases with m in the UB regime as m < N/2, exhibiting approximately a scaling variation V ∼ m^–z. On the other half portion m > N/2, the velocity increases in a mirror-symmetric way because it is a quasi-equilibrium process. The extracted exponent z is 1.83(3), 1.63(4), and 1.39(3) for the three studied salt cases, respectively. The results are consistent with the α_U exponent reported in Table 1 via the relation α_U = 1 + z. Here, is the deduction

6

As the mean number of the monomers in the cis region decreases with time, we should look at the curves from the right side. The V curve decreases first and turns to show increase in a process, while the V_e curve increases only with time since the very beginning. The difference shows the influence of the bouncing stage on the calculation of the mean drift velocity. The V_e curve joins the V curve later at some place, at about m < N/2. At the final moment near m = 1, the velocity V_e is curved down to become smaller than V. It is the small impact of the chain leaving stage.

The velocity curve V is raised up as the driving field enters the WD regime. A constant velocity portion, resulting from the flattened dwelling time W(s), appears on the middle of the curve. Near the end of the process (in the small m region), the V curve manages to rejoin the UB’s curve. Consequently, V follows the same scaling variation and ends up with a velocity profile similar to the one in the UB regime. In the SDT and SDI regimes, the driving field is sufficiently strong to influence the entire process. The process attains a fast velocity profile with the V curve lying fully above the UB’s one. Under this circumstance, the V_e curves show no difference from the V curves. A stronger driving field is required to bring the curves into this situation if the salt valence is higher. The plots also reveal that the velocity profile is relatively flat for the case of Z = 3.

3.5. Diffusion in Translocation

After having studied the drift velocity, we turn to investigate the associated property in translocation—diffusion. The diffusion property can be quantified by calculating the variance of the translocation coordinate. The variance, by definition ⟨Δs²(t)⟩ = ⟨(s(t) – ⟨s(t)⟩)²⟩, measures the spreading of the coordinate s(t) with respect to the mean value ⟨s(t)⟩ and is a function of time t. It generally scales with time as ⟨Δs²(t)⟩ ∼ t^γ. A normal diffusion behavior is characterized by the exponent γ equal to 1. For the situation γ < 1, the behavior is sub-diffusive. If γ is larger than one, it is super-diffusive.

Figure 13a,b presents the results of the two kinds of variance function in the trivalent salt solution. The first one, denoted by ⟨Δs²⟩, is calculated by skipping the bouncing stage where the average is started from the time point of the last bouncing event of each individual process. It is expected to give the true spreading information of translocation for real experiments. The second one, denoted by ⟨Δs²⟩_e, is calculated by setting the starting time of the average at the switching-on of the driving electric field (aka the starting of the process). The ⟨Δs²⟩_e function involves the contribution of the variance from the bouncing stage and can deviate significantly from the ⟨Δs²⟩ if E is small. The two kinds of function shall drop to zero at some moment when the chain completely leaves the cis region because the terminated s has a value N_m in each process, which leads to a null variance.

Figure 13.

Variances of translocation, (a) ⟨Δs²⟩ and (b) ⟨Δs²⟩_e, as a function of time t. The chain length N is 123. The strengths of the driving field are given in the legends.

We find that the scaling of ⟨Δs²⟩ starts with t⁰ and soon changes to t¹ in the UB and WD regimes. Near the end of a process, a small surge appears and thereafter, the curve drops quickly to zero. It shows that under the weak driving condition, the translocation is basically proceeded with a normal diffusion behavior. If the driving field is increased to be in the SDT or SDI regimes, the ⟨Δs²⟩ turns to follow a t² scaling. The diffusion behavior becomes a ballistic-type super-diffusion. It suggests a constant velocity profile, which is consistent with the observation of Figure 12c. Notably, the peak of the curve decreases its height with increasing E and tends to be a constant as the driving field is in the strongly driving condition. It ascertains a decreasing behavior of the maximum s-fluctuation with E.

The ⟨Δs²⟩_e curves are essentially identical to the ⟨Δs²⟩ in the WD, SDT, and SDI regimes (E ≥ 1.5) because the influence of the bouncing stage is negligible. In the UB regime, a sub-diffusive behavior ⟨Δs²⟩_e ∼ t^0.7 is observed, spanning in the early stages of the process. It results from the dominant contribution of the head monomer bouncing to overcome the translocation barrier.

To understand the influences of the salt valence on the diffusion property, we plot ⟨Δs²⟩ versus the scaled time t̃ ≡ t/⟨τ⟩ for Z = 1, 2, and 3, as in shown Figure 14. We find that the variances follow astonishingly a universal curve in the UB regime (refer to Figure 14a), although the processes did show different drift velocity profiles in the different salt solutions (refer to Figure 12). As E is increased, the collapsed ⟨Δs²⟩ curves separate, particularly visible in the WD regime, as shown in Figure 14b. The diffusion behavior switches gradually from a normal diffusion to a super diffusion. Figure 14c shows that the separation between the curves diminishes as the driving field enters the strongly driving regimes. It is because the trajectory of a translocation is more deterministic under the strong driving conditions and therefore, the uncertainty (or the spreading of the s state at a given time) is reduced. Under this condition, the diffusion is mainly a ballistic super-diffusion. It is worth noting that the ⟨Δs²⟩ curves are terminated at around the same place for different salt valences in the scaled time scale. The sharp termination shows that a majority of the translocation events are completed within the duration t̃ of about 3.4, 2.1, and 1.4, respectively, for the presented three driving fields. This information provides a useful estimate for the upper bound of the translocation time for an individual process with respect to the mean time ⟨τ⟩ at a given studied condition.

Figure 14.

Variance ⟨Δs²⟩ vs the scaled time t̃ at (a) E = 0.1, (b) E = 1.0, and (c) E = 64.0 in the three salt solutions. The value of N is 123.

4. Discussions and Conclusions

We have investigated forced translocation of polyelectrolytes across a membrane through a nanopore in a trivalent salt solution by means of Langevin dynamics simulations. A reflective wall boundary was introduced at the pore exit to prevent the falling-back of the chain into the cis region. This boundary setting is important for the feasibility of the simulations, especially when the applied driving field is weak. Using the setting, we were able to investigate forced translocation in a complete way, with the driving field E covering all levels of the forces from a negligibly weak to an extremely strong driving force condition. Two different translocation times were studied: the first one τ measures the duration since the occurrence of the last bouncing of the head monomer against the reflective wall until the first leaving of the tail monomer from the cis region, and the second one τ_e accounts for the duration of a process started at the switching-on of the driving field and ended at the final leaving of the chain from the cis region. No bouncing-related process is involved in the measuring of the time τ; therefore, τ is close to the translocation time observed in real experiments. The comparison between τ_e and τ allowed us to understand the impact of the introduced reflective wall and the diffusion characteristics of translocation.

The mean translocation time was studied under the scaling form ⟨τ⟩ ∼ N^αE^–δ. Four distinct variational behaviors have been identified as E was increased under the fixed N condition. It permitted us to classify the driving field strength into the four regimes: UB, WD, SDT, and SDI, similar to the ones classified in the monovalent and divalent salt solutions.⁷⁰ Using the newly defined translocation time, we were able to calculate correctly the δ-exponent in different force regimes: δ_U = 0, δ_T ≃ 1.6, and δ_I ≃ 1.0 for the trivalent salt case. Table 1 shows that the re-calculated exponents in the monovalent and divalent salt solutions have also the closed values. Noticeably, in the WD regime, only the monovalent salt case exhibited scaling variation, which gave δ_W ≃ 1.0. The nonlinear decondensation of the counter ions in the divalent and trivalent salt solutions deviated from the expected linear response of the threading behavior and as a result, δ_W cannot be defined in the two salt solutions. The study of the escape time showed that ⟨τ_e⟩ can be two orders of magnitude larger than ⟨τ⟩ in the UB regime. Therefore, it is important to use ⟨τ⟩, rather than ⟨τ_e⟩, in the study of the translocation behavior. We decomposed the time difference ⟨τ_e⟩ – ⟨τ⟩ into ⟨τ_b⟩ and and showed that the bouncing time ⟨τ_b⟩ exhibits four distinguishable behaviors with E. It alternatively gave clear evidence for the existence of the four nominated force regimes. The time measuring the required time for the chain tail to leave the cis region was found to be negligibly small and will not change the scaling behavior of translocation whether it is involved in ⟨τ⟩ or not.

The exponents α_U, α_T, and α_I were calculated by varying N at fixed E. Surprisingly, the value of α_U was found to decrease with increasing the salt valence. This counter-intuitive result came from the competition of the two opposite effects: (1) the presence of the multivalent counter ions collapses the chain into a compact structure, which increases the difficulty of threading, and thus the translocation time should be increased and (2) the collapse of the chain reduces the required displacement for the chain to diffuse across the membrane and hence the translocation time decreases. Our study showed that the second effect won the competition. The exponent α_W cannot be measured in the study because no clear scaling variation with N was observed under the corresponding weakly driving situation. The value of α_T was found around one and the value of α_I lay between 1.3 and 1.5.

The experiments have found that DNA translocation can be significantly slowed down by adding divalent salts in the solutions.²⁷⁻³¹ To compare the results, we plot the experimental translocation time versus the chain length in Figure 15, together with our simulation data converted to the real physical units, regardless of the differences in the settings of the pore size, the salt concentration, the ion type, the chain structure, the chain stiffness, the membrane thickness, and so on (refer to Supporting Information for details).

We discovered that all the experimental data lie in an extension area of the WD regime. It totally makes sense because for a practical reason, people would like the translocation velocity to be as slow as possible to increase the readability of the genetic codes. Therefore, the applied driving field should not be large. It is obviously not feasible to place the system in the UB regime because the bias is too weak to maintain a reasonable rate of translocation. Consequently, staying in the WD regime should be the optimal choice, as having been done in the experiments. According to our results, using trivalent salts in a translocation process is not as advantageous as we thought in slowing down the process. It did increase the required time for a chain to overcome the translocation barrier, but the increase in the proper translocation time was found to be limited. However, a nontrivial change in the order of the translocation time against salt valence was observed: the trivalent salt case has the smallest ⟨τ⟩ in the UB regime but turns to possess the longest time in the SDT regime as E increases. Therefore, it is possible to find an optimized condition of translocation to obtain the desired slowing-down effect in the WD regime. This possibility has been seen in Figure 4a where the trivalent salt displayed the largest ⟨τ⟩ in the WD regime among the three salt cases.

In addition to the translocation time, we went further to investigate variational behaviors of various important physical quantities in the four force regimes. For example, the radii of gyration of the chain portions on the two sides of the membrane (Figure 5) showed a time-reversible symmetry in the UB regime. It indicated that the translocation was a quasi-equilibrium process. Increasing E drove the system out of the quasi-equilibrium situation by skewing the curves to the left because the trivalent counter ions did not have enough time to condense back onto the chain in a fast translocation. The number of the condensed counter ions (Figure 8) was thus drastically reduced on the trans side, compared to the quasi-equilibrium case. As a result, the effective chain charge (Figure 9) became very negative, approaching the chain bare charge.

The calculation of the two waiting time functions, W(s) and W_e(s) (refer to Figure 10), allowed us to understand the dynamics of translocation: how the chain dwells at each translocation state s and how the entire curve is varied with the driving field. The W(s) function, which did not account for the head monomer bouncing and the tail monomer leaving, was found mirror symmetric to the middle of the process in the UB regime. It fulfilled the symmetric requirement for a quasi-equilibrium translocation. On the contrary, the W_e(s) waiting time function involved the full process and displayed a huge and broad peak at the beginning of translocation, showing the surmounted difficulty of threading in overcoming the entropic barrier. The combined function W_b(s) + (s), as shown in Figure 11 revealed the “spectrum” of the dwelling time of the chain in the bouncing and the leaving stages.

The “drift” of translocation (Figure 12) was studied by taking the reciprocal of the waiting time. The variation of the velocity V exhibited interesting scaling behaviors against the number m of monomers in the cis region under the UB condition. The extracted exponent z was shown to be related with α_U by the addition of the constant, one. A noticeable finding was that the velocity followed a common tread of profile near the end of a process when the driving field is weak. Under strong drivings, the velocity curves were moved upward to completely leave the common tread. We also calculated V_e for comparison to understand the impact of the reflection wall boundary.

The diffusion property (Figures 13 and 14) was investigated by calculating the variance of s. The results showed that the main process had ⟨Δs²⟩ following normal diffusion in the UB and WD regimes. It turned to exhibit super-diffusion behavior with γ ≃ 2 as the driving field became strong enough. The variance curves were terminated abruptly at about 3.4 ⟨τ⟩ in the UB regime and at about 1.4 ⟨τ⟩ in the strongly driving regimes. The sharp termination indicated that the translocation time of an individual process is bounded in the entire driving force space. All the above results have been compared with the cases in the monovalent and divalent salt solutions: the figures were plotted beside the ones of the trivalent salt and the influence of salt valence on the studied quantities can be visualized directly. The studied topics cover a wide extent of the details of translocation properties, including the threading time, the chain size, the ion condensation, and the drift and diffusion properties. It provides valuable information for researchers in the understanding of charged-chain translocation and in the development of nanopore sequencing techniques. In the future, free energy calculations will be performed to study nonequilibrium thermodynamics of the polyelectrolyte in a translocation process.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.0c02647.

• Translocation time in the monovalent and divalent salt solutions, variations of the exponents δ and α, and experimental data (PDF)

The author declares no competing financial interest.